Location and Mereology

Substantivalists believe that there are regions of space
or spacetime. Many substantivalists also believe that there are
entities (people, tables, electrons, fields, holes, events, tropes,
universals, …) that are located at regions. For these
philosophers, questions arise about the relationship between located
entities and the regions at which they are located. Are located
entities identical to their locations, as
supersubstantivalists maintain? Are they entirely separate from their
locations, in the sense that they share no parts with them?

Without yet taking a stance on these metaphysical questions, some
philosophers have tried to formulate a minimal core of
‘conceptual’ truths governing the location relation and
its interaction with parthood and other mereological relations. Two
overarching questions arise. In what ways (if any) must the
mereological structure of a located entity mirror the mereological
structure of its location? And in what ways (if any) must the
mereological relationships between some things mirror the mereological
relationships between the locations of those things?

Recent philosophical literature in this area focuses largely on
three questions, each corresponding to a different way in which the
relevant mirroring might fail:

Is interpenetration possible? That is, can
entities that do not share parts be exactly located in regions that
do share parts?

Are extended simples possible?

Is multilocation possible? That is, is it
possible for an entity to be exactly located in more than one
spacetime region, or in more than one region of space at the same
time?

The present article takes on these questions and addresses a number
of other issues along the way.

In keeping with the recent literature, this article will focus on
‘entity-to-region’ location relations—i.e., location
relations that paradigmatically hold between entities
and regions. We will ignore location relations that hold
between entities and what seem to be nonregions. (Consider ‘the
transmitter is located at the top of the ridge’. Prima
facie, neither the transmitter nor the top of the ridge is a
region.)

Since our focus is on entity-to-region location relations, we will
work under the following controversial but popular assumption:

(1)

Substantivalism
There are such entities as regions.

Unless otherwise noted, we will assume that the regions in question
are spacetime regions, where these are thought of in
accordance with the traditional ‘Block Universe’ picture,
as captured by the following:

(2)

Eternalism
The past, present, and future are all equally real.

(3)

The B-theory of Time
No time is present in any absolute, not-merely-indexical sense.

(4)

The Spacetime View There is just
one fundamental spatiotemporal arena: spacetime. Instants and
intervals of time, if there are such things, are just spacetime
regions of certain sorts. Likewise for points and regions of
space.

Relatively little depends upon (2)–(4), however. If one dislikes
talk of spacetime, one should be able reinterpret most of what we say
here in terms of regions of space without loss of plausibility.
Further, we assume that

(5)

Subregions as Parts If x
and y are both regions, then x is a subregion
of y if and only if x is a part of y.

(5) contrasts with another fairly common view according to which
regions are sets of points and have their subregions as subsets.

We make the following assumptions about parthood, where
‘x is a part of y’ is symbolized
‘P(x,y)’:

(6)

Reflexivity of Parthood
∀xP(x,x)Each thing is a
part of itself.

(7)

Transitivity of Parthood
∀x∀y∀z[[P(x,y)
& P(y,z)]
→ P(x,z)]If x is a part
of y and y is a part of z, then x is a
part of z.

We will also find it convenient to have predicates for overlapping,
disjointness, proper parthood, simplicity, complexity, gunkiness, and
mereological coincidence. We define them as follows:

Finally, we will treat the expressions ‘x is composed
of the ys’, ‘x is a sum of
the ys’, and, ‘x is a fusion of
the ys’ as meaning ‘each of the ys is a part
of x and each part of x overlaps at least one of
the ys.’

As the above remarks suggest, we will operate under the
assumptions

that there is just one fundamental parthood relation, and

that this relation holds both among regions (one region can be a
part of another region) and among entities that are located at
regions (one material object, e.g., can be a part of another
material object), and

that it is a two-place relation that does not ‘hold
relative to’ times, regions or sortals.

All these assumptions are controversial. Some philosophers have
explored the view that the fundamental parthood relation is
three-place (Thomson 1983; Hudson 2001; Balashov 2008; Donnelly 2010)
or four-place (Gilmore 2009; Kleinschmidt 2011) (for more on the
latter, see Section 3.2 of the supplementary document
Additional Arguments).
Others have
defended the view that one fundamental parthood relation holds between
regions, another between material objects (McDaniel 2004, 2009).

This entry addresses questions framed in terms of modal
notions. Are extended simples possible? Is it necessary that nothing
is multilocated? All talk of possibility and necessity in what follows
should be understood as talk of metaphysical possibility and
metaphysical necessity, respectively. In keeping with current
orthodoxy, we assume that being metaphysically necessary (a property
of propositions or sentences) is not to be identified with being a
logical truth, being an analytic truth, being a conceptual truth, or
being an a priori truth. Although metaphysical necessity is
not identified with conceptual truth—and, correlatively,
metaphysical possibility is not identified with
conceivability—one might still think that conceivability (or
something in that vicinity) provides prima facie evidence for
metaphysical possibility. I leave open the question of whether the
laws of nature are metaphysically necessary, but I take it that the
standard answer is ‘No’.

One final preliminary. For better or worse, the recent literature
on location and mereology tends to set aside complications arising
from vagueness and indeterminacy. We will do the same.

Debates about location have been framed in terms of two main
locational predicates: ‘is exactly located at’ and
‘is weakly located at’. Philosophers tend to agree about
how these predicates apply in particular cases, and they tend to agree
that one of these predicates should be taken as primitive and used to
define the other. But there is disagreement about which to define and
which to take as primitive.

Here is a typical intuitive gloss for ‘is exactly located
at’: a entity x is exactly located at a region y
if and only if x has (or has-at-y) exactly the same
shape and size as y and stands (or stands-at-y) in all
the same spatial or spatiotemporal relations to other entities as
does y.[1]
Thus, small cubes are exactly located only at small cubical regions,
large spheres are exactly located only at large spherical regions, and
so on (see Casati and Varzi 1999: 119–120; Bittner, Donnelly,
and Smith 2004; Gilmore 2006: 200–202; Sattig 2006: 48).

Here is a typical intuitive gloss for ‘is weakly located
at’: an entity x is weakly located at a region y
if and only if y is ‘not completely free
of’ x (Parsons 2007: 203). Thus I am weakly located at a
certain region, r, whose shape, size, and position perfectly
match my own, I am weakly located at the bottom half of that region, I
am weakly located at a certain much larger
region, r+, that has r as a proper part, and
I am weakly located at a certain scattered region, r*, that's
made up of the bottom half of r together with a small region
somewhere in Siberia. In short, if r is the one and only region
at which I am exactly located, then I am weakly located at just those
regions that overlap r.

Figure 1

Figure 1 illustrates exact location and weak location. The dotted
lines indicate regions
(r1–r5). The shaded circle
indicates a disc-shaped object, d, that is located at regions.
In the diagram, d is exactly located
at r4 but not
at r1, r2, r3,
or r5, and d is weakly located
at r2, r3, r4,
and r5 but not at r1.

Assuming that we now grasp the intended relations, we are in a
position to consider proposals about how the associated predicates
might be defined. Suppose that we take ‘x is exactly
located at y’ as primitive and symbolize it as
‘L(x, y)’ (Casati and Varzi
1999). Then, as Parsons (2007: 204) notes, the following definition of
‘is weakly located at’ (‘WKL’) is
natural:

an analytic truth (Parsons 2007). This might be seen as a drawback,
for the following reason among others (Gilmore 2006: 203; Parsons
2007: 207–9). One might think that the following situation is
possible:

all regions are extended and gunky and decompose into smaller
(but still extended and still gunky) regions,

some located entities are point-like and unextended, and

entities are located only at regions.

A point-like entity will be too small to be exactly located in any
extended region, but presumably it should still be weakly located at
many regions—viz., each in a series of nested regions that
‘converge’ on the point-like entity. Thus, if it's
possible that (i)–(iii) are all true, then it's possible that,
contrary to Exactness, a thing is weakly located somewhere without
being exactly located anywhere.

On the other hand, suppose that we take ‘WKL’ as
primitive. How might we use this predicate to define
‘L’? This is less obvious, but the definition that
Parsons (2007: 205) proposes is

(D9)

Exact LocationL(x, y)
=df∀z[WKL(x, z)
↔ O(y, z)]‘x is
exactly located at y’ means ‘x is weakly
located at all and only those entities that
overlap y’

To see how this definition works, return to the object d in
Figure 1. We want it to turn out that d is exactly located
at r4 and nowhere else. Given the natural
assumptions about which regions are parts of which, (D9) delivers the
desired verdicts.

For example, (D9) tells us that d is not exactly located
at r3, for the reason that r3
overlaps certain regions (e.g., r1) at
which d is not weakly located. In other
words, r3 ‘overlaps regions it
shouldn't’. The situation with r5 is
reversed. (D9) tells us that d is not exactly located
at r5, for the reason that there are regions at
which d is weakly located (e.g., r2)
that r5 does not overlap. In other
words, r5 ‘fails to overlap regions it
should’. Region r2 has both vices: it overlaps
certain regions it shouldn't (e.g., its southeast corner), and it
fails to overlap certain regions it should
(e.g., r5). Region r4 has neither
vice. Hence d counts as being exactly located at it, according
to (D9).

One potential problem with (D9), however, is that it makes the
following principle

an analytic truth. As we will see in
section 7,
there are many who would deny Quasi-functionality, and there
are others who deny that it is necessary, analytic, or in any sense
‘conceptual’.

In the spirit of van Inwagen (1981), one might take the fundamental
location relation to be lying-within, where this holds only between a
mereologically simple spacetime point, on the one hand, and a located
entity, on the other. One might then define exact location in terms of
it, as follows: ‘x is exactly located at y’
means ‘for any z, if z is simple, then: z
lies within x if and only if z is a part
of y’. This has consequences similar to those of (D3). It
has the additional disadvantage that, in the context of gunky
spacetime, it yields the result that everything is exactly located
everywhere.

Philosophers have put forward of variety of axiom systems that are
meant to capture the interaction between parthood and location. One
especially bold idea is that the mereological properties of, and
relations between, located entities perfectly match those of
their locations. This has been dubbed Mereological Harmony
(Schaffer 2009).

x is a proper part of y
iff x's location is a proper part of y's
location.

(H7)

x and y overlap iff x's
location and y's location overlap.

(H8)

The xs compose y iff the locations
of the xs compose y's location.

Some philosophers apparently take Mereological Harmony to be a
necessary truth (Schaffer 2009:
138).[2]
The remainder of this entry considers three
separate threats to the view that Mereological Harmony is necessary:
interpenetration (§4), extended
simples (§5), and multilocation
(§6).

(There are other threats to Mereological Harmony that we will not
discuss. We will not discuss threats to H7 and H8 that arise from
‘moderate views about receptacles’, according to which
only topologically open or topologically closed regions can be exact
locations (see Cartwright 1975; Hudson 2005: 47–56; and
especially Uzquiano 2006). Nor will we discuss the threats to H4
discussed in Uzquiano (2011).)

A case of interpenetration occurs when non-overlapping entities
have overlapping exact locations—e.g., when a ghost passes
through a wall. In such a case, the right-to-left direction of H7
fails. An extended simple is a simple entity with a complex exact
location: it violates the left-to-right direction of H1, the
right-to-left direction of the (equivalent) H2, and the left-to-right
direction of the instance of H3 that results letting n = 1. A
case of multilocation occurs when a given entity has more than one
exact location. This violates

which is left implicit in Saucedo's statement of Mereological
Harmony.

The three questions that we will consider—Is interpenetration
possible? Are extended simples possible? Is multilocation
possible?—are logically independent of one another. Thus there
is room for eight specific packages of views. See Figure 2.

Figure 2

Package

Interpenetration?

Extended simples?

Multilocation?

1

yes

yes

yes

2

yes

yes

no

3

yes

no

yes

4

yes

no

no

5

no

yes

yes

6

no

yes

no

7

no

no

yes

8

no

no

no

We conjecture that all but Package 5 have proponents, but we leave
it as an exercise to the reader to supply the relevant citations.

It is important to note that even if interpenetration, extended
simples, and multilocation are all possible, a number of substantive
and interesting principles linking parthood and location may still
survive. For example, the possibility of interpenetration and extended
simples poses no threat to (11) or (12):

(11)

Expansivity
⃞∀x∀y∀z∀w[[P(x, y)
& L(x, z) & L(y, w)]
→ P(z, w)]Necessarily, if x is a part of y, and if x is
exactly located at z and y is exactly located
at w, then z is a part of w: ‘the part's
location is a part of the whole's
location’.[3]

(12)

Delegation
⃞∀x∀y∀z[[C(x)
& L(x, y) & P(z, y)]
→ ∃w∃u[PP(w, x)
& O(u, y)
& L(w, u)]]Necessarily, if x is complex and is exactly located
at y, then for any part z of y, some proper
part w of x is weakly located at z, where
‘weakly located’ is defined via
(D2).[4]

Expansivity rules out cases like the following (Figure 3), in which
the object a is a part of the object o, but a's
exact location, ra, is not a part
of o's exact location, r.

Figure 3

The object a is part of the object o, but a's
exact location ra is not a port of o's
exact location, r.Ruled out by Expansitivity

The idea behind Delegation is that if x is complex, then if
you stick a pin into x's exact location, you will have stuck
that pin into the exact location of one of x's proper parts as
well. In slightly different terms, a complex entity can't be weakly
located at a certain region unless one of its proper parts—a
‘delegate’—is weakly located at that region as
well.

Delegation rules out cases like the following (Figure 4), in
which o* is a complex object that is exactly located at
region r*, but r* has a part ra
that does not overlap an exact location of any of o*'s proper
parts:

Figure 4

The region ra is a part of object o*'s exact location, and object o* is complex, but no proper part of o has an exact location that overlaps ra.Ruled out by Delegation

Neither interpenetration nor extended simples would threaten (11)
or (12). Would multilocation threaten them? Gilmore (2009) argues if
some entities are multilocated, then, in order to allow things to have
different parts at different locations, the fundamental parthood
relation will need to be a four-place relation expressed by
‘x at y is a part of z
at w’. Similar ideas are independently developed in
Kleinschmidt (2011). Given the four-place view, it is unclear how to
understand the two-place mereological predicates that (11) and (12)
employ. However, Gilmore formulates close analogues of (11) and (12)
in terms of a four-place parthood predicate. The four-place analogues
would appear to be unthreatened by interpenetration, extended simples,
or multilocation. See Section 3.2 of the supplementary document
Additional Arguments
for more on the four-place view.

According to No Interpenetration, it is metaphysically impossible
for entities of any type to ‘pass through one another’
without sharing parts—in the manner of a ghost passing through a
solid brick wall. In other words, it is impossible for entities to
‘interpenetrate’.

There is a closely related principle that deserves some
comment. Roughly put, the related principle says that, necessarily,
if x's exact location is a part of y's exact location,
then x is a part of y. In symbols:

(14)

⃞∀x∀y∀z∀w[(L(x, z)
& L(y, w)
& P(z, w))
→ P(x, y)]

This principle may seem to say basically the same thing as (13) but
to say it more simply—using the primitive predicate
‘P’ instead of the defined predicate
‘O’. Why then focus on (13) instead of (14)?

The reason for this is that some of the opposition to (14) will
stem from opposition to a purely mereological principle: Strong
Supplementation. It says that if every part of x
overlaps y, then x is a part of y. Those who deny
this will be very likely to deny (14), but they might still be
attracted to (13). Let Lump1 be a lump of clay, and let Goliath be the
statue that is ‘made out of’ Lump1. One might think that
every part of Goliath overlaps Lump1 (and vice versa), and that
Goliath's exact location is a part of (indeed, identical to) Lump1's
exact location, but that Goliath is not a part of Lump1 (Lowe
2003). In that case one will reject (14). But one might still go on to
say that ‘ghostly-interpenetration-without-part-sharing’
is impossible, and hence that (13) is true. After all, Goliath and
Lump1 share parts, as do Goliath's top two-thirds and Lump1's bottom
two-thirds, and so on.

In general, our task here is to set aside the purely mereological
controversies (see the entry on
mereology) and to focus instead on the issues that
are exclusively concerned with location and its interaction with
parthood. Too much of the controversy over (14) arises from
controversy over ‘pure mereology’. By contrast, if (13) is
controversial, this is only because of what it says about
the connections between parthood and location.

Immanent realists say that a universal is in some sense
‘wholly present’ in each thing that instantiates it
(Bigelow 1988; O'Leary-Hawthorne and Cover 1998; Paul 2002; Newman
2002). This makes it natural to hold that disjoint universals
frequently interpenetrate.

To see why, let e be an electron, and suppose that it
instantiates two different universals: a certain maximally determinate
mass universal, um, and a certain maximally
determinate charge universal, uc. Further,
suppose that e is exactly located at region r. Then it
will be natural for the immanent realist to say that

um is exactly located r as well,
or at least at some region rm that
has r as a part, and

uc is exactly located at r or
some region rc that has r as a
part.

(If these universals are also instantiated elsewhere, then it will
be debatable as to whether they are exactly located
at r. Perhaps um has only one exact
location—the sum of the exactly locations of its
instances. Likewise for uc.) Either way, the
immanent realist will say that um
and uc have exact locations that overlap. But
presumably um and uc
do not themselves overlap. Universals such as
these—perfectly natural, maximally determinate, non-structural,
non-conjunctive, etc.—are typically taken to be simple,
in which case they overlap only if they're identical, which they are
not. A similar point can be made in terms
of tropes—particular, spatiotemporally located
‘cases’ of properties or relations (Schaffer 2001). Mass
tropes and charges tropes frequently interpenetrate.

Three responses to this argument are worth considering.

The first response says: so much the worse for tropes and immanent
universals. This response uses a mereo-locational principle, No
Interpenetration, as a premise in an argument against certain
metaphysical views, namely those that posit tropes or immanent
universals. Is there some reason why mereo-locational principles
should not be used in this way? The principles of pure mereology are
often so used. For example, David Lewis (1999: 108–110) rejects
states of affairs and structural universals on the grounds that they
would violate the Uniqueness of Composition, the principle that
no xs have more than one mereological
sum.[5] Why not
give the same status to certain mereo-locational principles? Why not,
for example, say that No Interpenetration is better justified than is
the view that universals or tropes are spatiotemporally located? True,
it is hard to say much by way of argument for No
Interpenetration. But the same can be said of various purely
mereological principles, such as the Transitivity of Parthood, and
this is often treated as a non-negotiable constraint that any tenable
metaphysical position must obey.

The second response says that while immanent universals and/or
tropes are spatiotemporal entities that are ‘in their
instances’, they are not exactly located
anywhere. David Armstrong writes that

Talk of the location of universals, while better than placing
them in another realm, is also not quite appropriate… To talk
of locating universals in space-time then emerges as a crude way of
speaking. Space-time is not a box into which universals are
put. Universals are constituents of states of affairs. Space-time is
a conjunction of states of affairs. In that sense, universals are
“in” spacetime. But they are in it as helping to
constitute it. (1989: 99)

Simplified somewhat, the response holds that (i) universals are
parts or constituents of entities that have exact locations, and in
that sense they are ‘in their instances’, but (ii)
universals do no themselves have exact locations and hence do not have
overlapping exact locations. Given (ii), the universals or tropes in
question no longer count as examples of interpenetration. Call this
the Burying Strategy, since it ‘buries’
universals and/or tropes in located entities, rather than treating
them as being located.

Here is an especially vivid instance of the Burying Strategy:

All tropes are instantaneous, spatially
point-sized, and mereologically simple. All spacetime
points are instantaneous, spatially point-sized, and
mereologically complex. In particular, each spacetime point is a
fusion of some tropes (each of which is at zero distance from each
of the rest), and each trope is a part of exactly one spacetime
point. Something counts as a spacetime region if
and only if it is either a spacetime point or a fusion of some
spacetime points. Each spacetime region is exactly located at itself
and nothing else has an exact location.

On this view, tropes are parts of spacetime points and
regions but they do not themselves have exact locations at all. Hence,
even if two tropes are both parts of the same spacetime point, they do
not interpenetrate, in our sense.

The third response to the argument from universals and tropes is to
say, ‘True, universals and/or tropes can interpenetrate, but
material objects can't’. This grants the argument and rejects No
Interpenetration in favor of a weaker, restricted principle. If we
introduce a one-place predicate, ‘M’, for ‘is
a material object’, then we can state the restricted principle
as

Some think that it is possible for two disjoint material
objects to have overlapping exact locations. Perhaps there are no
actual cases of the relevant sort. Indeed, perhaps such cases are
physically impossible—ruled out by the laws of nature (though
see the next section). But one might still think that these cases are
metaphysically possible.

After all, what is it that keeps material objects from
interpenetrating in the actual world? Repulsive forces,
presumably. But a standard view is that the laws governing such forces
are not metaphysically
necessary.[6]
And on that assumption it is natural to
conclude that there are metaphysically possible worlds in which any
repulsive forces that exist can be overridden in such a way as to
allow material objects to interpenetrate (for more this, see Zimmerman
1996a and Sider 2000).

A similar line of thought is sometimes framed as a conceivability
argument. One might take cases of interpenetration to be
‘conceivable’ or ‘intuitively possible’, and
one might take this to be some prima facie evidence for their
possibility. In New Essays the Human Understanding
(II.xxvii.1), Leibniz writes that

we find that two shadows or two rays of light interpenetrate, and
we could devise an imaginary world where bodies did the same. (1704
[1996])

David Sanford describes such a scenario in more detail:

What imaginable circumstances would tempt us to say that two
[disjoint] things were in the same place at the same time? Imagine
two rectangular blocks of the same size and shape each moving along
a straight path perpendicular to the other. The blocks approach the
intersection of the paths at the same time, and apparently neither
slows down nor changes its direction. They seem to pass right
through one another, and they do this without changing with respect
to colour, texture, density, etc. We want to say that we have the
same two blocks with which we started. And we do not want to say
that either block passed out of existence and was shortly thereafter
re-created. Thus we want to say that each block moved along its path
without any spatio-temporal discontinuity. And we can say this only
if we admit that parts of one block simultaneously occupied the same
space as parts of the other block. (1967: 37)

An objection suggests itself. Why think that, in this scenario, the
blocks really are disjoint while they are passing through one another?
Why not say instead that they overlap?

In response, one might just stipulate that what is under
consideration are two always-disjoint blocks, and then insist that the
case remains conceivable, so described.

Alternatively, one might further specify the case in such a way as
to be able to argue for the blocks' disjointness. Consider a
certain region, r, that is said by Sanford to the exact
location of two disjoint material objects: p1 (a
part of Block 1) and p2 (a part of Block 2). One
might add that two incompatible properties are instantiated
at r—say, having mass of 2 kg,
and having mass of 3 kg—where each of these
properties is such that if it is instantiated by an entity x,
then it is instantiated by anything that mereologically coincides
with x. Presumably this is no less conceivable (or
‘intuitively possible’ or whatever) than Sanford's
original case. But in this new version of the case, it is not open to
Sanford's opponent to claim that the co-located objects are identical
or mereologically coincident with each other. For the region contains
a 2 kg object and a 3 kg object, and no one object is identical to (or
mereologically coincident with) both a 2 kg object and a 3 kg
object.

One question that has been raised in the recent philosophical
literature is whether contemporary physics provides us with examples
of disjoint fundamental particles that have the same, or overlapping,
exact locations. John Hawthorne and Gabriel Uzquiano apparently claim
that the answer is ‘Yes’. They write that

particles having integral spin—otherwise known as
bosons—in modern particle physics … are generally
thought to be point-sized. Moreover, according to the spin
statistics theorem, while fermions—point-particles with half
integer spin—cannot be colocated, bosons are perfectly well
able to cohabit a single spacetime point. (2011: 3–4)

Jonathan Schaffer suggests otherwise:

[a] more sophisticated treatment of these cases involves field
theory. Instead of there being two bosons co-located at
region r, there is a bosonic field with doubled intensity
at r. (2009: 140)

Whereas Hawthorne and Uzquiano apparently take bosons to
provide actual examples of interpenetration, Kris McDaniel
suggests that they at least reinforce the
conceivability—and prima facie possibility—of
such counterexamples:

what this example shows is that [disjoint] co-located material
objects are not merely conceivable, but that a tremendously detailed
conception of them has been formed: [disjoint] co-located objects play
a role in the interpretation of certain physical theories. It might be
that at the end of the day speculative physics will postulate
co-located material objects. It seems to me that we should not
disregard this possibility a priori. (2007a: 240)

If one's goal, in constructing a theory of location, is to
articulate the necessary and a priori truths governing
location and its interaction with parthood, then even McDaniel's
modest point still counts against including No Interpenetration in
one's theory. For if McDaniel is right, then that principle is at best
an a posteriori truth. (See Simons (1994, 2004) for further
discussion of bosons and for related considerations in support of
interpenetration.)

Theodore Sider (2000: 585–6), Kris McDaniel (2007a), and Raul
Saucedo (2011) have all objected to No Interpenetration on the grounds
that it conflicts with plausible ‘principles of
recombination’. McDaniel's formulation of the objection, which
is similar to Sider's, runs as follows:

The state of affairs in which [a simple] object x occupies
a particular region of space R (at t) is distinct from
the state of affairs in which [some other simple] object y
occupies the same region (at the same time). From the fact that the
first state of affairs obtains, we can infer nothing about the
location of y. Both states of affairs obtain contingently. If
any recombination of distinct, contingent states of affairs yields a
genuine possibility, as I am inclined to hold, then there are
possible worlds at which both x and y occupy R
(at t). (2007a: 241)

Let o1 and o2 be two different
objects, let r be a region, and consider the following states
of affairs:

s1

o1's being simple and exactly located at r

s2

o2's being simple and exactly located at r

Then we can reconstruct McDaniel's argument as follows:

Premise 1

s1 is a contingent state of affairs.

Premise 2

s2 is a contingent state of affairs.

Premise 3

s1 is distinct from s2.

Premise 4

For any x and any y, if x and y
are each contingent states of affairs, and if they are distinct
from each other, then possibly, both x and y
obtain.

Therefore

Conclusion

Possibly, both s1 and s2 obtain.

If it's possible for both s1
and s2 to obtain, then it's possible for a given
region to be the exact location of two different simples. And since no
two simples can overlap, this would mean that it's possible for
disjoint things (the simples) to have identical (hence overlapping)
exact locations.

Is the argument successful? As Sider and McDaniel are well aware,
the term ‘distinct from’ needs to mean something other
than ‘not identical to’, if the recombination principle,
Premise 4, is to get off the ground. After all, the state of affairs
of os being 2 kg in mass and the state of affairs
of os being 3 kg in mass are both contingent, and they
are not identical to each other, but presumably it is not possible
that they both obtain. (Presumably it's not possible that
something is both 2 kg and 3 kg in mass!)

But it is no easy matter to give ‘distinct from’ a
meaning that makes Premises 3 and 4 simultaneously plausible. If it
means ‘shares no parts or constituents with’, then Premise
4 avoids the counterexample given above, but Premise 3 ceases to be
plausible, since s1 and s2 do
plausibly share a constituent, namely r. If ‘s is
distinct from s*’ is defined as ‘(i)
possibly, s obtains and s* does not, (ii)
possibly, s does not obtain and s does, (iii) possibly,
neither s nor s* obtains, and (iv) possibly,
both s and s* obtain’, then Premise 4 is trivially
true, but Premise 3 begs the question.

For additional arguments in favor of interpenetration, including a
more detailed recombination argument,
see Section 1 of the supplementary document
Additional Arguments.

Supersubstantivalism is often glossed as the view that each
material object is identical to some spacetime region at which the
object is exactly
located.[7]
If supersubstantivalism is not just true
but necessary, then it is impossible for disjoint material
objects to be exactly located at overlapping regions. After all,
if material object x is exactly located at
region r1 and material object y is exactly
located at r2, and r1
and r2 overlap, then, given
supersubstantivalism, x=r1
and y=r2, and hence x and y
overlap as well.

This gives us an argument for (13*), the version of No
Interpenetration restricted to material objects. But it does not give
us an argument for (13), the full-strength version of No
Interpenetration. For it wouldn't follow that it is impossible for
there to be disjoint entities of any kind to have overlapping
exact locations. Supersubstantivalism, as stated above, leaves open
the possibility that there are universals or tropes, e.g., that are
exactly located at regions but not identical to them. Hence it leaves
open the possibility that disjoint entities that are not material
objects might have overlapping exact locations.

To construct an argument for No Interpenetration, we would need to
appeal to a stronger supersubstantivalist doctrine, for example:

(15)

Supersubstantivalism+
⃞∀x∀y[L(x, y) → x=y]Necessarily, each entity is identical to anything at which it is
exactly located.

Supersubstantivalism+ entails No Interpenetration. Take any
objects x and y in any possible world, and suppose that
they have exact locations, r1
and r2 respectively, that overlap. Then, given
Supersubstantivalism+, x=r1
and y=r2, hence x and y
overlap.

For a second, related argument against interpenetration,
see Section 1.2 of the supplementary document
Additional Arguments.

A simple is an entity that has no proper parts. Are there any
simples? Within the realm of spatiotemporal entities, some natural
candidates are: spacetime points, fundamental particles such as
electrons (or instantaneous temporal parts of them), and perhaps
certain universals, certain tropes, and/or certain sets. On the other
hand, it would seem to be an empirically open possibility that all
spatiotemporal entities are gunky.

Say that an entity is extended just in case it is a
spatiotemporal entity and does not have the shape and size of a
point. In this sense of ‘extended’, a solid cube would
count as extended, but so would the mereological sum of two
point-particles that are one foot apart. Although such a sum would
have zero length, it would be a scattered object and so would
not have the shape of a point.

Are there any extended simples? Could there be? Those who answer
‘No’ to both questions will be inclined to accept

(16)

No Extended Simples (NXS)
⃞∀x∀y[[L(x, y)
& C(y)]
→ C(x)]Necessarily, if x is
exactly located at y and y is complex, then x is
complex.

Strictly speaking, NXS does not say that extended simples are
impossible; rather, it says that simples with complex exact locations
are impossible. It leaves open the possibility that there are extended
simple regions and extended simple entities that are exactly located
at them (for more on this topic, see Tognazzini 2006, Braddon-Mitchell
and Miller 2006, Spencer 2010, and Dainton 2010: 294–301). And
it rules out the possibility that there is a point-sized material
simple that is exactly located at a point-sized but mereologically
complex region (e.g., a region that is the fusion of several
point-sized tropes each of which is at zero distance from each of the
others).

For the most part, however, it will do no harm to treat the debate
over extended simples as a debate over NXS. We can do so if we assume
that, necessarily, a region is extended if and only if it is
complex. So, in what follows, we will operate under that assumption
unless we explicitly note otherwise.

An initial argument appeals to the claim that extended simples are
conceivable and takes that to be some prima facie evidence in
favor of their possibility. To conceive of an extended simple, think
of an extended—say, cubical—object that has no proper
parts. The idea is not, or not merely, that the cube cannot be
physically split or cut up. Whether or not it can be split is a
separate question. The idea is that there are no proper parts of the
cube. Although the object is cubical, it has no top half or bottom
half, no left half or right half. All there is, in the relevant
vicinity, is the cube itself—together with the region at which
it is exactly located and the parts of that region.

Since this argument does not appear to raise any concerns that are
specific to extended simples, we will move on without further
comment.

Debates about extended simples typically focus on the question of
whether a material object could be an extended simple. But one option
is to say that, whether or not material objects can be extended
simples, there are entities in some other ontological category
(universals, tropes, sets) that can be. Here is an argument from
immanent universals:

(17)

My body instantiates the universal being 70 kg in mass.

(18)

The universal being 70 kg in mass is simple.

(19)

My body is exactly located at a complex, spatially extended
spacetime region.

(20)

For any x, any y, and any z, if x
instantiates y and x is exactly located at z,
then y is exactly located at z or at some region of
which z is a part.

(21)

For any z and any z*, if z* is a region
that has a complex, spatially extended spacetime region as a part,
then z* is itself complex and spatially extended.

Therefore

(22)

There is a simple entity (being 70 kg in
mass) that is exactly located at a complex, spatially
extended spacetime region—either my body's exact location,
or some larger region that has that location as a part.

If one thinks that universals have exact locations, one will say
that a universal is exactly located either:

at each region at which one of its instances is exactly located,
or

at the region that is the sum of the exact locations of its
instances.

Those who opt for (i) will say that being 70 kg in
mass has the same exact location as my body. Those who opt for
(ii) will say that being 70 kg in mass has a
vastly larger exact location—namely, the sum of the
exact locations of all of the 70-kg entities in the universe. Either
way, if the given universal is a simple, it will count as an extended
simple. (A parallel argument from tropes is also available.)

If one is convinced that extended simples are impossible but
insists on treating universals as both simple and located in
spacetime, one could say that universals are multilocated rather than
extended. In particular, one could reject options (i) and (ii) above
in favor of the view that a universal u is exactly located
at r if and only if r is a simple spacetime point that
is a part of some exact location of some instance of u. Of
course, not everyone finds multilocation any more palatable than
extended simples, but some do (see Hudson 2001, 2005).

Finally, one might think that universals, tropes, or sets can be
extended simples, but that material objects cannot. In that case one
will reject NXS in favor of a weaker principle that is restricted to
material objects:

What are strings made of? There are two possible answers to this
question. First, strings are truly fundamental—they are
“atoms,” uncuttable constituents. … As the
absolute smallest constituents of anything and everything, they
represent the end of the line … in the numerous layers of
substructure in the microscopic world. From this perspective, even
though strings have spatial extent, the question of their
composition is without any content. Were strings to be made of
something smaller they would not be fundamental. Instead, whatever
strings were composed of would immediately displace them and lay
claim to being an even more basic constituent of the universe
… [A] string is simply a string—as there is nothing
more fundamental, it can't be described as being composed of any
other substance. (1999: 141–2)

Can strings be treated as being identical to the spacetime regions
at which they are exactly located? Greene does not explicitly address
this question. If the answer is ‘Yes’, however, and if
strings are exactly located only at complex regions, then string
theory would not be committed to extended simples after all.

As we noted in section 4.4, Sider,
McDaniel, and Saucedo have argued for the possibility of
interpenetration by appeal to principles of recombination. They have
also marshaled these principles in support of the possibility of
extended simples (Sider 2007; McDaniel 2007b; Saucedo 2011). Here is
the core of McDaniel's argument:

(NNC): Let F and G be accidental, intrinsic
properties; let R be a fundamental relation; let x
and y be contingently existing non-overlapping entities. Then
it is not the case that, necessarily, Rxy only
if (Fx if and only if Gy)…if we
accept the Humean premise that there are no necessary connections
between the accidental, intrinsic properties of regions of space and
the accidental, intrinsic properties of material objects, then we
should hold that there are no necessary connections between the
mereological structure of a material object and the mereological
structure of the region it occupies. Specifically, it is not true
that, necessarily, a material object is a simple if and only if it
occupies a simple (read: pointsized) region of spacetime. It follows
that extended material simples are possible. (2007b:
135–137)

If we let o and r be contingently existing,
non-overlapping entities, then McDaniel's argument can be
reconstructed as follows:

For any accidental, intrinsic properties F
and G, any fundamental relation R, and any
contingently existing non-overlapping entities x
and y, it is not the case that, necessarily if Rxy
then (if Fx
then Gy).[8]

Therefore

(28)

It is not the case that necessarily, if o is exactly
located at r, then: if o is simple then r is
a simple region.

Supersubstantivalists will deny that exact location is a
fundamental relation. Others might take aim at McDaniel's
recombination principle. Let F be the
property being round and let G be the
property not being square. Prima facie, these would
seem to be accidental and intrinsic. But if they are, and if exact
location is fundamental, then McDaniel's principle tells us that it is
possible for a round object to be exactly located at a square region,
which some might take as a reductio of (27).

For further arguments in favor of extended simples, including a
more detailed recombination argument,
see Section 2.1 of the supplementary document
Additional Arguments.

According to supersubstantivalism, each material object is
identical to the spacetime region at which it is exactly
located. Those who take supersubstantivalism to be a necessary truth
are likely to be hostile to the possibility of extended
simples—or at least to the possibility of extended simple
material objects. For they are likely to endorse the following
argument:

(29)

Necessarily, all material objects are regions.

(30)

Necessarily, all extended regions are complex.

Therefore

(31)

Necessarily, all extended material objects are complex. In
other words, it is impossible for a material object to be an
extended simple.

Most defenders of extended simples will resist this argument by
denying (29). They tend to think of material objects as being
non-identical to, and indeed mereologically disjoint from, the regions
at which they are exactly located (Markosian 1998; McDaniel 2007b,
2009; Parsons 2007; Saucedo 2011). A different option, however, is to
reject (30) in favor of the view that there are, or at least could be,
extended simple spacetime regions (Braddon-Mitchell and Miller 2006;
Tognazzini 2006; for arguments to the contrary, see Spencer 2010).

The above argument yields only a weak, restricted version of NXS,
the ban on extended simples. It leaves open the possibility of
extended simples that are not material objects.

NXS does follow, however, from Supersubstantivalism+, according to
which it's a necessary truth that each entity is identical to anything
at which it is exactly located. For suppose that x is exactly
located at y and that y is complex. Then, given
Supersubstantivalism+, x=y, hence x is complex
too.

A second argument against extended simples arises from problems
about simples whose intrinsic properties vary across space or
spacetime:

(32)

If it is possible for there to be an extended simple, then it
is possible for there to be an extended simple that exhibits
intrinsic qualitative variation across spacetime—e.g. an
extended simple that is white in one region and grey in
another. (Colors are just placeholders; presumably more realistic
examples are available.)

(33)

Necessarily, for any x, if x varies across
spacetime with respect to its intrinsic properties, then there are
incompatible intrinsic properties F and G and
parts x1 and x2 of x
such that x1 instantiates F
and x2 instantiates G.

(34)

Necessarily, for any x, if there are incompatible
intrinsic properties F and G and
parts x1 and x2 of x
such that x1 instantiates F
and x2 instantiates G, then x is
not simple.

Therefore

(35)

It is not possible for there to be an extended simple.

The argument is valid and (34) is clearly true. If x has a
part that instantiates F and a part that instantiates G,
then, provided that these properties are incompatible, the parts in
question are not identical. So x has at least two parts. So at
least one of them must be a proper part of x, in which
case x is not simple.

One might resist the argument by denying (32), that is, by saying
that a simple can be extended but only if it is qualitatively
homogeneous across spacetime. Most friends of extended simples,
however, have apparently granted (32) and rejected (33) instead. This
can be done in at least four ways.

5.6.1 The relativizing approach

The relativizing approach says that if a simple, o, is
white in one region and grey in another, then o bears the
being white at relation to the first region and the being grey at
relation to the second; it need not have a part that instantiates
the property whiteness or a part that instantiates the property
greyness. Alternatively, one might treat instantiation as a
three-place relation, and say that o instantiates the
property whiteness at one region and instantiates the property
greyness at another.

Both versions of this approach are based on theories of
persistence and change that attempt to reconcile eternalism and
the B-theory of time with endurantism, the view that objects are
wholly present at multiple times and do not have temporal parts
(for more on endurantism, see
section 6.3).
Accordingly, the standard criticisms of the relevant
theories of persistence and change (on which see Haslanger 2003)
would seem to apply equally to the relativizing approach to the
problem of qualitative variation for extended simples. The main
criticism is that, by appealing to relations to regions, the
relativizing approach makes an object's color distribution (or
whatever) a dyadic relation or an extrinsic property, when it is
in fact (says the critic) an intrinsic property, involving only
the object, its parts, and the intrinsic properties of these
things.

5.6.2 The stuff-theoretic approach

Ned Markosian (1998, 2004a) responds to the problem of
qualitative variation by appeal to the distinction between things
(e.g., material objects) and stuff (e.g., matter). Stuff,
according to Markosian, comes in portions, but no portion of stuff
is a thing. Both things and portions of stuff can have parts, but
any part of a thing is itself a thing, and any part of a portion
of stuff is itself a portion of stuff. Nevertheless, things and
portions of stuff do stand in an important relation: each thing
is constituted by a portion of stuff. With this framework
in place, Markosian offers the following account. There can be
extended simple things, but there cannot be extended simple
portions of stuff. When an extended simple, o, is white in
region rw and grey in
region rg, this isn't because o
itself has a part that is exactly located
at rw and instantiates whiteness and
another part that is exactly located
at rg and instantiates greyness; rather,
it's because the portion of stuff that constitutes o has
such parts. This preserves the simplicity of the extended thing in
question. It may also avoid the complaint, facing the relativizing
approach, that it makes the color distribution of an extended
simple an extrinsic matter. The relation between a thing and the
portion of stuff that constitutes it is apparently more intimate
than the relation between a thing and the region at which it is
exactly located. Moreover, the stuff-theoretic approach, unlike
the relativizing approach, holds that when an extended object is
white in one region and grey in another, the monadic,
intrinsic properties whiteness and greyness (not merely
the dyadic relations being white at
and being grey at) are both
instantiated. However, some may doubt the coherence of a notion of
constitution that allows a complex entity to constitute a simple
one.

5.6.3 Distributional properties

Josh Parsons (2000) proposes that if a simple is white in one
region and grey in another, then it has a fundamental,
intrinsic, distributional property. Some distributional
properties, such as being jet black all over,
are uniform. Others, such as being
polka-dotted or being striped,
are non-uniform. When a simple has a non-uniform
distributional property, this is not grounded in its having parts,
configured in a certain way, that each have simpler, uniform
properties. Nor is it grounded in the simple's standing in
incompatible relations (being white at
and being grey at) to different spacetime
regions. Rather, it is a basic, ungrounded fact about the simple
in question. This apparently avoids the worries about
extrinsicness that face the relativizing approach, and it makes no
appeal to Markosian's unorthodox notion of constitution.

As McDaniel (2009) notes, however, Parsons is apparently forced to
treat facts of the form x is F at r as
unanalyzed and ungrounded, which one might take to be a drawback. In
Figure 5, the objects o1 and o2
presumably have the same maximally determinate distributional color
property: roughly, being grey in one half and white in
the other.

Figure 5

Call this property P. Further,
object o1 is grey at
region r1g. What grounds this fact? Not
the fact that o1 has P. After
all, o2 also has P, but it is not grey
at r1g. Not the fact
that o1 has P and is exactly located
at r1. For it is possible that there be a thing
that has P and is exactly located
at r1g but is not grey
at r1g: this is the case, e.g., in the
possible world that results from deleting o1 and
putting o2 in its place (without rotating it
clockwise or counterclockwise). Thus, in addition to his basic
distributional properties, Parsons appears forced to treat
relations to regions (e.g., being grey at) as
being basic as well.

The relativizing approach has no such problem: it treats
relations to regions as basic and analyzes distributional
properties in terms of them. The stuff-theoretic approach avoids
the problem too. It analyzes facts of the form x
is F at y as: either x or the portion of
stuff that constitutes x has a part that is exactly located
at y and is F.

5.6.4 Tropes

McDaniel (2009) attempts to avoid all the above problems by
appeal to a theory of tropes developed by Ehring (1997a,b). The
idea is that a simple can be grey in one region and white in
another region by instantiating a greyness trope that is exactly
located at the first region and instantiating a whiteness trope
that is exactly located at the second region. This permits an
analysis of facts of the form x is F
at y as x instantiates an F-ness
trope that is exactly located at y; it makes no appeal
to stuff or portions of stuff; and, since it
invokes monadic color tropes, it (arguably) refrains from
treating the facts about a simple's color distribution as being
fundamentally relational and extrinsic.

(For further elaboration and defense of the argument from
qualitative variation, see Spencer (2010).)

For further arguments against extended simples,
see Section 2.2 of the supplementary document
Additional Arguments.

To say that an object is multilocated is to say that it has more
than one exact location: ‘x is multilocated’ means
‘∃y1∃y2[L(x, y1)
& L(x, y2)
& y1≠y2]’. We
consider a series of putative examples of multi-location
in section 6.3.

Opponents of multilocation accept Functionality+. Friends of
multilocation typically want to affirm something stronger than the
negation of Functionality+. They typically accept the possibility of
an entity that is exactly located at each of two regions that do not
even overlap.

Earlier we glossed ‘x is exactly located
at y’ as ‘x has (or has-at-y) the
same size and shape as y, and stands (or stands-at-y) in
all the same spatiotemporal relations to things as
does y’. Thus spheres are exactly located only at
spherical regions, cubes only at cubical regions, and so on. When an
entity is said to be multilocated, then, it is said to stand in this
relation to each of several regions: informally put, it has
the same size, shape, and position as region r1; it
has the same size, shape, and position as region r2;
and so on. No claim is made to the effect that the object is exactly
located at the sum
of r1, r2, …, or at
any proper parts of any of these regions.

To clarify the idea of multilocation in an informal way, it may be
useful to consider Figure 6, inspired by Hudson (2005: 105).

Figure 6

A scattered, singly located object

A non-scattered, multilocated object

The object o1 is scattered: its shape is that of
the sum of two non-overlapping circles. It is not
multilocated. Rather, it has just one exact location: the scattered
region r3. It is not exactly located at any proper
part of that region, such as r1
or r2.

The object o2 is multilocated. It has two (and
only two) exact locations. It is exactly located at the circular
region r3; and it is exactly located at the circular
region r4, which does not
overlap r3. It is not exactly located at their sum,
and it is not located at any of their proper
parts. Since o2 is exactly located
at r3, which is circular, o2 is
circular, at least at r3. For parallel
reasons, o2 is circular at r4. By
contrast, o1 is not circular simpliciter,
nor is it circular at any region.

It is natural to think that if these two objects
were visible, they would be visually
indistinguishable. Indeed, it is tempting to think that there would be
no empirically significant difference between o1
and o2. For those with verificationist leanings,
this may lead to the belief that there is no qualitative difference at
all between o1 and o2 and hence
that there must be something defective about the initial set-up of the
case. This line of thought seems to lie behind much of the hostility
toward multilocation, though it has rarely if ever been developed in
any detail.

As with interpenetration and extended simples, one might offer a
conceivability argument for the possibility of multilocation. One
could claim that multilocation is conceivable (alternatively:
intuitively possible) and take this to be some prima facie
evidence that multilocation is possible. Since this argument does not
appear to raise any issues that are specific to multilocation, we will
move on.

It is natural to think that the extant recombination arguments for
interpenetration and extended simples can be adapted to yield an
argument for multilocation. If one slightly alters the passage (quoted
in section 4.4) in which McDaniel presents
his recombination argument for interpenetration (2007a: 241), one gets
the following, parallel argument for multilocation:

The actual state of affairs in which an object x is exactly
located at a particular region r is distinct from the merely
possible state of affairs in which object x is exactly located
at some other, disjoint region r*. From the fact that x
is exactly located at r, we can infer nothing about what, if
anything, is exactly located at r*. Both states of affairs are
contingent: they possibly obtain and possibly fail to obtain. If any
recombination of contingent states of affairs yields a genuine
possibility, then there are possible worlds at which x is
exactly located both at r and at r* and hence is
multilocated.

This argument would seem to have the same virtues and vices as
McDaniel's original argument for interpenetration.

(Surprisingly, however, some recombination arguments for
interpenetration and extended simples cannot be adapted to
yield an argument for
multilocation. See Section 3.1 of the supplemetary document
Additional Arguments.)

A third kind of argument for the possibility of multilocation
simply points to examples. Entities that have been taken to be
multilocated include: immanent universals, enduring material objects,
enduring tropes, four-dimensional perduring objects, backward time
travelers, fission products, transworld individuals, works of music,
and an omnipresent God.

6.3.1 Immanent universals

As we have noted, immanent realists say that universals are
spatiotemporal entities that are in some sense ‘wholly
present’ in the things that instantiate them. This seems to be
the view under discussion in the following passage from
Plato's Parmenides, 131a–131b (in Hamilton and Cairns
1961: 925):

Parmenides

Do you hold, then that the form as a whole, a single thing, is
in each of the many, or how?

Socrates

Why should it not be in each, Parmenides?

Parmenides

If so, a form which is one and the same will be at the same
time, as a whole, in a number of things which are separate, and
consequently will be separate from itself.

One natural way to translate immanent realism into the terminology
of exact location is via the following principle:

(37)

Necessarily, for any x, any y, and any z,
if x is exactly located at y and x
instantiates z, then z is exactly located
at y.

To see how this leads to multilocation, suppose that some monadic
universal u is instantiated by an entity e1
that is exactly located at region r1 and by a
different entity, e2, that is exactly located at
region r2, disjoint from r1. Then,
given (37), u itself is exactly located both
at r1 and at r2.

(37) is not inevitable, even for immanent realists. Some of them
might prefer to say that a monadic universal is exactly located only
at the sum of the exact locations of its instances (Bigelow,
1988: 18–27, can in places be read as embracing this). On this
view, a simple monadic universal might be scattered but would not be
multilocated. Others (Armstrong 1989: 99) prefer to say that
universals do not have exact locations at all, though they are parts
or constituents of things that have exact locations or of spacetime
itself. This was dubbed the ‘Burying Strategy’
in section 4.1.[9]

6.3.2 Enduring material objects

The debate over persistence through time centers around two rival
views, endurantism and perdurantism. Endurantists often say that a
persisting material object is temporally unextended and in some sense
‘wholly present’ at each instant of its
career. Perdurantists often say that a persisting material object is a
temporally extended entity that has a different temporal part
at each different instant of its career and hence is at most partially
present at any one instant.

(Informally, an instantaneous temporal part of me is an object that
is a part of me, is made of the exactly same matter as I am whenever
it exists, and has exactly the same spatial location as I do whenever
it exists, but exists at only a single
instant.[10])

Recently, a number of philosophers have suggested that the
traditional endurantism versus perdurantism dispute runs together a
pair of independent disputes about persistence: a mereological dispute
concerning the existence of temporal parts, and a locational dispute
concerning exact locations (Gilmore 2006, 2008; Hawthorne 2006; Sattig
2006; Eddon 2010; Donnelly 2010, 2011b; Rychter 2011). Stated somewhat
loosely, the mereological dispute is between the following views:

Mereological perdurance:
there are persisting material objects, and each such object has a
different temporal part at each different instant at which the
object exists.

Mereological endurance:
there are persisting material objects, but none of them has a
different temporal part at each different instant of its
career. (Perhaps none of them have any instantaneous temporal
parts—or any temporal parts aside from themselves—at
all!)

To frame the locational dispute, it will be useful to have one
further piece of terminology. Say that y is the path
of x if and only if y is the mereological sum of the
exact locations of x (Gilmore 2006: 204). (Strictly, we should
say ‘a path’ and ‘a sum’, unless we want to
presuppose Uniqueness: any xs have at most one sum.)
Informally, an object's path is the region that exactly contains the
object's complete career or life-history; it is the region that the
object exactly ‘sweeps out’ over the course of its
career.

We can then state the locational dispute as follows:

Locational perdurance:
there are persisting material objects, and each of them has one and
only one exact location—its path.

Locational endurance:
there are persisting material objects, and each of them has many
different exact locations, each such location being instantaneous or
‘spacelike’. Typically, each of these exact locations
will count as an instantaneous temporal part of the object's
path.

Philosophers on both sides of this dispute can agree about which
spacetime region is the path of a given material object. They will
disagree about which spacetime regions are the exact locations of the
object. The locational perdurantist will say that the object's only
exact location is its path. The locational endurantist will say that
the object is exactly located at many regions, each of them a slice of
its path. The interaction between the two disputes about persistence
is summarized in Figure 7 (from Gilmore 2008: 1230).

Figure 7

Locational endurance entails multilocation: it has it that some
material objects are exactly located at many different
regions. Importantly, however, mereological endurance, which
merely rejects temporal parts, does not entail multilocation. Thus one
might reject temporal parts while retaining Functionality. This is the
position of Parsons (2000, 2007). It corresponds to the lower
left-hand box in Figure 7.[11]

6.3.3 Enduring tropes

Whether or not one takes material objects to be multilocated in the
manner of locational endurantism, one might hold such a view about
tropes. Douglas Ehring (1997b, 2011) offers a detailed defense of the
view that tropes endure, though he does not say whether he takes them
to be multilocated in our sense. Consider some mereologically simple,
spatially unextended trope e, and suppose, with Ehring, that it
persists. Let the temporally extended, one-dimensional spacetime
region r be e's path. Then there are at least two
natural things one could say regarding e's locations. One could
say that (i) e has just one exact location, r itself, in
which case e is a temporally extended simple but not
multilocated. Alternatively, one could say (ii) e is exactly
located at each point in r (but nowhere else), in which
case e is multilocated but not extended.

6.3.4 Multilocated 4D worms

Hud Hudson (2001) defends a version of mereological perdurantism
according to which each ordinary material object is exactly located at
a great many (mostly overlapping) four-dimensional spacetime
regions. Hudson offers his view as a solution to ‘the problem of
the many,’ among other things.

(Hudson also argues that, given his view, the fundamental parthood
relation should be taken to be a three-place relation holding between
two objects and a region. This aspect of Hudson's theory is discussed
in detail by Gilmore (2009) and especially Donnelly (2010).)

6.3.5 Backward Time Travelers

Suppose that Suzy is born at time t0 and dies 90
years later, at time t90. At
time t40, the forty-year-old Suzy steps into a time
machine and disappears, reappearing (with her time machine) out of
thin air many years earlier. Suzy then enters the nursery in the house
where she was raised and, at t1, sees a one-year-old
baby, whom she knows to be herself. (The character Suzy originates in
Vihvelin 1996.)

If one is an endurantist, one will find it natural to describe this
situation as involving multi-location in space or spacetime: Suzy is
exactly located at rA, a region with the size
and shape of an adult; Baby Suzy is exactly located
at rB, a disjoint but simultaneous region
with the size and shape of a baby; and Suzy=Baby Suzy. Hence there is
a single thing (namely, Suzy/Baby Suzy) that is exactly located at two
different, simultaneous regions, rA
and rB (Keller and Nelson 2001; Gilmore 2003,
2006, 2007; Miller 2006; Carroll 2011; Kleinschmidt 2011). Moreover,
all of this holds even if one rejects talk of regions
of spacetime in favor of talk of regions
of space.

To be sure, even endurantists can deny that the situation involves
multilocation. Those who accept mereological endurantism and
locational perdurantism will say that Suzy does not have temporal
parts but is still four-dimensional in at least the following sense:
she is exactly located at only one region: her four-dimensional path
(Parsons 2000, 2007, 2008). If one is a presentist—one
who says that only the present time and its contents exist—one
might say that, at t1, Suzy is exactly located only
at the sum of rA
and rB, rather than at either of those
regions themselves (Markosian 2004c).

However, if Suzy is exactly located only at the sum
of rA and rB
at t1 then she is not shaped like a human being
at t1; rather, she is a scattered object shaped like
the sum of two human beings. It might seem doubtful that Suzy can
exist at a time without being even approximately human-shaped at that
time. This would speak in favor of the view that Suzy is multilocated,
rather than scattered,
at t1.[12]

6.3.6 Fission

Barry Dainton (2008: 364–408) argues that if a person were to
undergo fission (in roughly the manner of an amoeba), the person would
continue to exist after the fission in bilocated form: she would be
wholly present in two different places at once (this view is also
discussed by Johnston 1989). Dainton's view is naturally understood as
involving multilocation, either in spacetime or in space at a time:
after the fission, the person is exactly located at two different
regions (see Gilmore, 2008: 1246, for an objection to Dainton's
view).

6.3.7 Modal Realism with Overlap

McDaniel (2004) develops a version of concrete modal realism that
posits multilocation; he dubs the view Modal Realism with
Overlap (MRO). Like Lewis's (1986) modal realism, MRO treats
possible worlds as concrete, spatiotemporal entities and holds that
spacetime regions are ‘worldbound’: no spacetime region
that is part of one world overlaps any spacetime region that is part
of another world. Unlike Lewis's view, however, MRO claims that
material objects are wholly present in more than one
world. Specifically, MRO entails multilocation. It says that if Lewis
is a philosopher in world w1 and a plumber but not a
philosopher in world w2, then he is exactly located
at two or more regions (one of which is a part of w1
and another of which is a part of w2).

6.3.8 Works of Music

Chris Tillman (2011) develops a view about the ontology of musical
works according to which Beethoven's 9th Symphony
(the symphony itself, as opposed to any particular performance of it)
is multilocated. The view has it that a work of music is exactly
located at a spacetime region r if and only if some performance
of that work is exactly located at r. When combined with facts
about the relevant performances, this principle yields the result that
Beethoven's 9th is exactly located at spacetime regions
confined to the 19th century Vienna and also at other
regions confined to 20th century New York; it is
multilocated. Tillman argues that this view is preferable to other
forms of ‘musical materialism’. He does not say whether it
is preferable to the view that works of music are abstract objects
with no spatiotemporal location at all.

6.3.9 God

Hudson (2009) notes that one might interpret the doctrine of divine
omnipresence as entailing that God is exactly located at each region
of spacetime. (For a discussion of other religious views to which
multilocation is relevant see Hudson (2010) and Effingham (forthcoming.)

As we noted in section 2, if one
defines ‘x is exactly located at y’ as
‘x is weakly located at all and only those entities that
overlap y’—as Parsons (2007) claims that we are
free to do—then one can argue as follows:

(38)

Necessarily, for any x and any y, x is
exactly located at y if and only if for any y*, x
is weakly located at y* if and only y overlaps y*
(by the definition of ‘is exactly located at’).

Therefore

(39)

So, necessarily, for any x, any y1, and
any y2, if x is exactly located
at y1 and x is exactly located
at y2, then y1 mereologically
coincides with y2 (from (38)).

(40)

Necessarily, for any y1 and
any y2, if something is exactly located
at y1 and something is exactly located
at y2 and y1 mereologically
coincides with y2,
then y1=y2. (Uniqueness for
Locations)

Therefore

(41)

Necessarily, for any x, and y1 and
any y2, if x is exactly located
at y1 and x is exactly located
at y2,
then y1=y2 (from (39) and
(40).)

To see that the inference from (38) to (39) is valid, suppose that
object o is exactly located at
regions ra
and rb. Since o is exactly located
at ra, o is (by (38)) weakly located
at all and only the entities that
overlap ra. Likewise, since o is
exactly located at rb, o is weakly
located at all and only the entities that
overlap rb. So ra
overlaps a given entity if and only if o is weakly located at
that entity; and rb overlaps a given entity
if and only if o is weakly located at that
entity. Hence ra
and rb overlap exactly the same entities: in
other words, they mereologically coincide. The rest of the argument is
self-explanatory.

The argument may persuade some. However, those who are initially
inclined to take the possibility of multilocation seriously may see
this argument as a reason to doubt the first premise and the
associated definition (Gilmore 2006: 203). (A parallel argument runs
through an alternative definition: ‘x is exactly located
at y’ means ‘for any z, if z is
simple, then z lies within x if and only if z is
a part of y’. As we noted in
section 1, this definition has the additional vice that in the context of
gunky spacetime, it yields the result that everything is exactly
located at every region.)

When taken to be a necessary truth, supersubstantivalism yields
arguments against the possibility of interpenetrating material objects
and extended simple material objects. Given the symmetry and
transitivity of identity, it also yields an argument against the
possibility of multilocated material objects:

(42)

Necessarily, if x is a material object and is exactly
located at y, then x=y.

Therefore

(43)

Necessarily, if x is a material object and is exactly
located at y1 and at y2,
then y1=y2.

Informally, the idea is this: since, necessarily, material objects
are identical to their exact locations, and since nothing can be
identical with two different entities, no material objects can have
two different exact locations.

This leaves open the possibility of multilocation for entities
(e.g., universals) that are not material objects. This possibility is
ruled out by Supersubstantivalism+, according to which it is necessary
that any entity is identical to anything at which it is exactly
located. Neither material objects nor universals can
be identical to more than one thing.

Extended simples face a problem arising from qualitative
variation. Multilocated entities face a similar problem. One might
think that if multilocation is possible, then there should be a
possible situation involving a thing o1 exactly
located at a region r1 and a
thing o2 exactly located at a disjoint
region r2, such that o1 is
white, o2 is grey,
and o1=o2. (Again, change in color
is just a placeholder; any apparently intrinsic qualitative change
would suffice for the purpose of this argument.) Since it's impossible
for a single thing to be both white and grey, one might continue, no
such situation is possible. This might lead one to conclude that
multilocation itself is impossible. In numbered form, we have:

(44)

If it is possible for there to be a multilocated entity, then it
is possible for there to be objects x1
and x2 and disjoint regions y1
and y2 such that: x1 is exactly
located at y1, x2 is exactly
located at y2, x1 is
white, x2 is grey,
and x1=x2.

(45)

Necessarily, nothing is both white and grey.

Therefore

(46)

It is not possible for there to be a multilocated entity.

Friends of multilocation reject the first premise, though not
always for the same reason. Some say that (i) multilocation is
possible but only for universals or tropes, and that (ii) these
entities do not vary between locations with respect to anything in the
vicinity of their ‘intrinsic properties’, such as shape,
size, mass, or color.

Others say that multilocation is possible even for entities
that do undergo such variation, such as material objects. In
that case, one can adopt a relativizing approach to the
so-called ‘intrinsic properties’ with respect to which the
given things can vary: one can say that object x1
(i.e., x2) stands in the being white
at relation to region y1 and
the being grey at relation to
region y2 (van Inwagen 1990a). Thus, in the
situation described above, x1 is neither white nor
grey. It is white at region y1 and grey at
region y2. The relations in question are
incompatible, in the sense that no one thing can bear both of them to
the same entity, but it is possible for a thing to bear one of them to
one region and the other to a different region.

It is an interesting question to what extent the friend of
multilocation can invoke other responses to the problem of change. Can
she appeal to stuff, distributional properties, or short-lived tropes,
as did the friend of extended simples, in response to the problem of
qualitative
variation?[13]

For further arguments against multilocation,
see Section 3.2 of the supplementary document
Additional Arguments.

There are some important questions concerning parthood and location
about which we have so far said little or nothing. These include the
following:

What topological constraints are there (if any) on the sorts of
regions that can be the exact locations of material objects? For
example, is it a necessary truth that material objects are exactly
located only at

three-dimensional regions?

topologically closed regions, regions with an
‘outermost skin’ of points?

–––, 2009, “Why Parthood Might Be a Four
Place Relation, and How it Behaves if it Is,” in L. Honnefelder,
E. Runggaldier, and B. Schick, eds., Unity and Time in
Metaphysics (Berlin: de Gruyter, 2009), pp. 83–133.

Smith, N. J. J., 2005, “A plea for things that are not quite
all there: Or, Is there a problem about vague composition and vague
identity?”, Journal of Philosophy, 102:
381–421.

Smith, S., 2007, “Continuous Bodies, Impenetrability, and
Contact Interactions: The View from the Applied Mathematics of
Continuum Mechanics”, British Journal for the Philosophy of
Science, 58: 503–538.

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